Question

A box contains 5 red and 7 white balls. Two balls are drawn at random. What is the probability that both are red?

Hint:

When objects are drawn {without replacement}, use combinations to count selections. For selecting \(r\) objects from \(n\): \[ \binom{n}{r}=\frac{n!}{r!(n-r)!} \]

Answer 1

Concept: Probability of an event is given by \[ P(E)=\frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}} \] When selecting objects without replacement, combinations are often used to count the possible selections.
Step 1: Find the total number of balls. Number of red balls = 5 Number of white balls = 7 Total balls: \[ 5+7=12 \]
Step 2: Find total ways to select 2 balls from 12. \[ \text{Total outcomes} = \binom{12}{2} \] \[ = \frac{12\times11}{2}=66 \]
Step 3: Find favourable outcomes (both red). Number of ways to select 2 red balls from 5: \[ \binom{5}{2} \] \[ =\frac{5\times4}{2}=10 \]
Step 4: Calculate the probability. \[ P(\text{both red})=\frac{\binom{5}{2}}{\binom{12}{2}} \] \[ =\frac{10}{66} \] \[ =\frac{5}{33} \]
Step 5: Final Answer. \[ \boxed{P(\text{both balls are red})=\frac{5}{33}} \]